Fischer-Clifford matrices and character tables of some group extensions

Abstract

A useful way for studying the properties of a group G is to express the elements of G in terms of matrices or permutations [27]. In 1878 Cayley showed that every group G is isomorphic to a subgroup of the symmetric group Sa, where Sa is the group of all permutations on G. A representation of a group G is a homomorphism T: G-+ GL(n, F) from G into the group GL(n, F) of n x n invertible matrices over a field F. The character of G afforded by a representation T is the trace of the matrices T(g) for each g E G. The table of characters of G is called the character table of G [20]. Since the completion of the classification of finite simple groups in 1981 [4], current research work in group theory involves the study of the structures of simple groups. The structures and character tables of maximal subgroups of simple groups give substantive information about these groups. Most of the maximal subgroups of simple groups [8] and some of their constituent groups are of extension type (i.e. a group G = N.G such that N <I G and G/N ð€€ƒ G). In our research we are particularly interested in faithful permutation representa­tions of sporadic simple groups and their automorphism groups [27]. The Mathieu groups are examples of sporadic simple groups [32]. A permutation group G on X is said to be k-transitive on X if for any two k-tuples (x1, x2, ... , xk) and (y1, Y2, ... , Yk) of k distinct elements of X, there exists g E G such that xf = Yi, 1 :S i :S k. Apart from the symmetric groups Sn and Alternating groups An, the Mathieu groups are the only non-trivial faithful k-transitive permutation groups for k = 4, 5 [32]. Now as can be seen from the Atlas of Finite Groups [8], the Mathieu group M22 has a maximal subgroup of form 24 : S5. Likewise the Mathieu group M23 has a maximal subgroup of form 24 :A7. As part of this project we will study the groups of forms 24 : Ss and 24 : A7. We note that some groups of form pn -l : Sn, where p is prime, have been studied in [35], however the group 24 : Ss studied here is not one of those groups. This will give some information about the structures of these groups. Let m, n E N, the set of positive integers and Zm = {O, 1, ... , m - 1} be the set of residues modulo m, also considered as a cyclic group Cm of order m. One of the groups to be studied in this work is a subgroup of the symmetric group Bmn of degree m x n. For example, we know from [28] that the simplectic group SP(6, 2), which is a maximal subgroup of the Fischer group Fi22 , has a subgroup of form 25 : 85 . Further the group 25 : S5 has a subgroup isomorphic to the split extension (S3)2 : C2 or the wreath product of the symmetric group S3 of degree 3 with the cyclic group C2 of order 2. In this project, we will also study the groups (Sn)m : Cm, where mis prime and n is a positive integer. The group (Sn)m : Cm is a subgroup of the group Smn of degree m x n. Several methods for constructing the character tables of group extensions exist. However, Fischer [12] has given an effective method for constructing the character tables of some group extensions including the groups cited above. This method known as the technique of the Fischer-Clifford matrices makes use of Clifford Theory [7, 20]. Given a group extension G = H.G such that every irreducible character of H can be extended to its inertia group, for each conjugacy class of G we construct a matrix called a Fischer-Clifford matrix of G. By using the Fischer-Clifford matrices of G together with the fusion maps and the character tables of inertia factor groups of G, we are able to construct the character table of G. We note that Fischer-Clifford matrices satisfy certain properties which may be used to construct them. The method of the Fischer-Clifford matrices has been used in many works both on split and non-split extensions [1, 10, 12, 25, 28, 30, 34, 35]. Here we will use the method of Fischer-Clifford matrices to construct the character tables of the groups 24 : S5, 24 : A1, (S3)2 : C2, (S3)3 : C3, (S4)2 : C2 and (